Caching Dynamic Skyline Queries D. Sacharidis 1 , P. Bouros 1 , T. Sellis 1,2 1 National Technical University of Athens 2 Institute for Management of Information Systems – R.C. Athena July 11 SSDBM'08
Outline • Introduction – Skyline (SL) and dynamic skyline queries (DSL) • Related work • Evaluating dynamic skyline queries – Computing orthant skylines (OSL) – Computing dynamic skyline via caching • LRU, LFU, LPP cache replacement policies • Experimental evaluation • Conclusions and Future work July 11 SSDBM'08
Skyline queries (SL) • Given a dataset of d- dimensional points – SL contains points not dominated by others – x dominates y iff x as good as y in all dimensions and strictly better in at least one July 11 SSDBM'08
Skyline queries (SL) • Given a dataset of d- dimensional points – SL contains points not dominated by others – x dominates y iff x as good as y in all Price dimensions and strictly Distance from sea better in at least one • Example – Dataset of hotels – Prefer cheap hotels close to the sea July 11 SSDBM'08
Skyline queries (SL) • Given a dataset of d- dimensional points – SL contains points not Skyline points dominated by others – x dominates y iff x as good as y in all Price dimensions and strictly Distance from sea better in at least one • Example – Dataset of hotels – Prefer cheap hotels close to the sea July 11 SSDBM'08
Skyline queries (SL) • Given a dataset of d- dimensional points – SL contains points not Skyline points dominated by others – x dominates y iff x as good as y in all Price p 1 dimensions and strictly Distance from sea better in at least one • Example – Dataset of hotels – Prefer cheap hotels close to the sea July 11 SSDBM'08
Skyline queries (SL) • Given a dataset of d- dimensional points – SL contains points not Skyline points dominated by others – x dominates y iff x as good as y in all Price p 1 p 2 dimensions and strictly Distance from sea better in at least one • Example – Dataset of hotels – Prefer cheap hotels close to the sea July 11 SSDBM'08
Dynamic skyline queries (DSL) • Extension of skyline queries – Given a query point q – DSL contains points not dynamically dominated by others w.r.t q – x dynamically dominates y iff x as preferable as y w.r.t. q in all dimensions and strictly more preferable w.r.t. q in at least one • Can be treated as static SL – Transform points w.r.t. q July 11 SSDBM'08
Dynamic skyline queries (DSL) • Extension of skyline queries – Given a query point q – DSL contains points not Query point q dynamically dominated by others w.r.t q – x dynamically dominates y Price iff x as preferable as y w.r.t. q in all dimensions and Distance from sea strictly more preferable • Example w.r.t. q in at least one • Can be treated as static – User defines “ideal” SL hotel q – Transform points w.r.t. q July 11 SSDBM'08
Dynamic skyline queries (DSL) • Extension of skyline queries – Given a query point q Dynamic – DSL contains points not Skyline points dynamically dominated by others w.r.t q q – x dynamically dominates y Price iff x as preferable as y w.r.t. q in all dimensions and Distance from sea strictly more preferable • Example w.r.t. q in at least one • Can be treated as static – User defines “ideal” SL hotel q – Transform points w.r.t. q July 11 SSDBM'08
Dynamic skyline queries (DSL) • Extension of skyline queries p 4 – Given a query point q p 5 Dynamic – DSL contains points not Skyline points dynamically dominated by others w.r.t q q – x dynamically dominates y Price iff x as preferable as y w.r.t. q in all dimensions and Distance from sea strictly more preferable • Example w.r.t. q in at least one • Can be treated as static – User defines “ideal” SL hotel q – Transform points w.r.t. q July 11 SSDBM'08
Intuition (1) • Traditional SL algorithms need to run anew for each DSL query • Our idea – Exploit results from past queries to reduce processing cost for future DSL queries – Cache past queries – Decide which queries in cache are useful July 11 SSDBM'08
Intuition (2) Price Distance from sea July 11 SSDBM'08
Intuition (2) • 2 past DSL queries – q a , q b • Each query partitions q a space in 4 quadrants q b Price Distance from sea July 11 SSDBM'08
Intuition (3) p 4 • A new query q arrives p 2 • Consider DSL for q a p 1 p 3 – p 1 is contained DSL(q a ) q a – p 1 dominates p 2 , p 3 , p 4 • p 1 lies in upper right q quadrant w.r.t. q a Price q b • q a lies in upper right quadrant w.r.t. q Distance from sea • p 1 dominates also p 2 , p 3 , • Shaded area denotes p 4 w.r.t. to q points dominated by p 1 – Exclude p 2 , p 3 , p 4 from dominance test for DSL(q) July 11 SSDBM'08
Contribution in brief • Caching past DSL queries cannot reduce processing cost for future ones – We need more information about dominance relationships • Introduce orthant skylines (OSL) and examine their relationship with DSL • Extend Bitmap algorithm to compute OSL in parallel with DSL • Cache OSL to enhance DSL queries evaluation – Present 3 cache replacement policies • LRU, LFU, LPP • Experimental evaluation of caching mechanism July 11 SSDBM'08
Related work • Non-indexed methods – Block-Nested Loops (BnL) – Bitmap – Multidimensional Divide and Conquer (DnC) – Sort First Scan (SFS) • Index-based methods – B-tree • sort points according to the lowest valued coordinate – R-tree • Nearest neighbor based (NN) • Branch and bound (BBS) July 11 SSDBM'08
Related work • Non-indexed methods – Block-Nested Loops (BnL) – Bitmap – Multidimensional Divide and Conquer (DnC) – Sort First Scan (SFS) • Index-based methods – B-tree • sort points according to the lowest valued coordinate – R-tree • Nearest neighbor based (NN) • Branch and bound (BBS) July 11 SSDBM'08
Bitmap • BnL variant • Suitable for domains with low cardinality and discrete • In brief – Computes a bitmap representation of the points in the dataset – Examines each point separately (dominance test) • Checks whether it is contained in the skyline or not • Exploits fast bitwise operations OR/AND July 11 SSDBM'08
Bitmap – Dominance test • For each point p – Define A = A 1 & A 2 & … & A d • Denotes the points as good as p in all dimensions – Define B = B 1 | B 2 | … | B d • Denotes the points strictly better than p in at least one dimension – Dominance test: • If C = A & B has all bits set to 0 then p is in SL July 11 SSDBM'08
Orthant skyline (OSL) • OSL provides more information about dominance relationships than DSL – Useful for pruning • Given a dataset of d- dimensional points and a query point q – Space partitioned in 2 d orthants – o-th orthant skyline (OSL) of q contains points of the o-th orthant not dynamically dominated by others inside orthant o w.r.t q July 11 SSDBM'08
Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL – Useful for pruning • Given a dataset of d- Query point q dimensional points and a query point q – Space partitioned in 2 d Price orthants – o-th orthant skyline (OSL) of q Distance from sea Quadrant 2 contains points of the o-th Quadrant 3 orthant not dynamically dominated by others inside orthant o w.r.t q July 11 SSDBM'08
Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL – Useful for pruning • Given a dataset of d- Query point q dimensional points and a query point q – Space partitioned in 2 d Price orthants – o-th orthant skyline (OSL) of q Distance from sea Quadrant 2 contains points of the o-th Quadrant 3 orthant not dynamically dominated by others inside orthant o w.r.t q July 11 SSDBM'08
Orthant skyline (OSL) Quadrant 1 Quadrant 0 • OSL provides more information about dominance relationships than DSL – Useful for pruning • Given a dataset of d- Query point q dimensional points and a query point q Quadrant 2 skyline points – Space partitioned in 2 d Price orthants – o-th orthant skyline (OSL) of q Distance from sea Quadrant 2 contains points of the o-th Quadrant 3 orthant not dynamically dominated by others inside orthant o w.r.t q July 11 SSDBM'08
OSL and DSL relationship Quadrant 1 Quadrant 0 q Price Distance from sea Quadrant 2 Quadrant 3 July 11 SSDBM'08
OSL and DSL relationship Quadrant 1 Quadrant 0 q Price Distance from sea Quadrant 2 Quadrant 3 July 11 SSDBM'08
OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 q Price Distance from sea Quadrant 2 Quadrant 3 July 11 SSDBM'08
OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q q Price Distance from sea Quadrant 2 Quadrant 3 July 11 SSDBM'08
OSL and DSL relationship Quadrant 1 Quadrant 0 • Map points from quadrants 1,2,3 to points inside quadrant 0 • Compute DSL w.r.t. q q • Union of all OSLs is Price superset of DSL w.r.t. Distance from sea Quadrant 2 to q Quadrant 3 July 11 SSDBM'08
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